Shohei pitching in relief and batting

OK so I just heard the conversation in the game about Shohei playing as a relief pitcher and I'm confused. Why can't the Dodgers do a 'double switch' and have him pitch and change from DH to Pitcher. So that now they have no DH and it would be like old school NL lineups with a pitchers spot and whoever is the new pitcher would have to bat? Or is that what would happen and I'm confusing myself?